Question

While working on her bike, Amanita turns it upside down and givesthe front wheel a counterclockwise spin. It spins at approximatelyconstant speed for a few seconds. During this portion of themotion, she records the x and y positions and velocities, as wellas the angular position and angular velocity, for the point on therim designated by the yellow-orange dot in the figure.(Figure1)
Figure 1
Let the origin of the coordinate system be at the center of thewheel, the positive x direction to the right, the positive yposition up, and the positive angular position counterclockwise.The graphs (figure 2) begin when the point if at the indicatedposition. One graph may be the correct answer to more than onepart


1. Which of the graphs corresponds to x position versustime?

2. Which of the graphs corresponds to angular position versustime?

3. Which of the graphs corresponds to y velocity versustime?

4. Which of the following graphs corresponds to angular velocityversus time?

Answer 1

Concepts and reason

This question is based on the concept of the periodic motion.

Periodic motion is a motion that repeats itself after a fixed time interval. They are characterized by periodic functions like $\sin$, $\cos$ etc. The motion of the mark along both $x{\rm{ }}$and $y$axis is periodic. The mark repeats its $x{\rm{ }}$position and $y$position after each complete rotation.

Fundamentals

Angular velocity is defined as the rate of change of angular displacement.

$\omega = \frac{{{\rm{Angular displacement}}}}{{{\rm{time}}}}$

When an object undergoes circular motion with constant speed, the entire motion is periodic. The object returns to its starting position after every full rotation. The position of object along $x{\rm{ }}$and $y$axis is also periodic since the $x{\rm{ }}$position and $y$position of the object repeats itself after each complete rotation around center.

(a)

Coordinate system be at the center of the wheel, the positive $x{\rm{ }}$direction to the right, the positive $y$position up and the positive angular position counterclockwise. The speed of rotation is constant. The motion along $x{\rm{ }}$axis is periodic and should be described by a periodic function. At $t = 0$, the point is at maximum position along $x{\rm{ }}$axis. The graph at $t = 0$ should also have positive value for $x{\rm{ }}$position.

The graph $F$representing $x$position versus time.

(b)

The angular position for a body stars rest with time is,

$\theta = \omega t$

Angular velocity $\omega$ remains same.

$\theta \propto t$

The graph $A$ representing angular position versus time.

(c)

The motion along $y$axis is also periodic and should be described by a periodic function.

At $t = 0$, the mark is at $y = 0$. The graph at $t = 0$ should be zero for $y$position.

After $t = 0$, the mark increases its $y$position till it reaches the top. The motion is up and down. the velocity verse time is simple harmonic. The velocity is maximum at mean position and zero at extreme position.

(d)

Velocity $v$ is constant during observation. Angular velocity $\omega$ is related to linear velocity $v$ and radius $r$ is written as,

$\omega = \frac{v}{r}$

Angular velocity is also constant. The wheel is spinning anticlockwise. The angular displacement is positive and increases continuously at a constant rate given by angular velocity. The graph should be a straight line with positive slope. Angular velocity is constant over time since velocity is constant.

Ans: Part a

The graph $F$representing $x$position versus time.